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3-manifolds abelian abelian group abstract algebra algorithm analysis analytic analytic functions applications approximation asymptotic August 23 automorphism Banach space boundary bounded California coefficients cohomology compact complex computation congruence conjecture construction continuous functions convergence convex countable curvature defined denote differential equations dimension domain elements equivalent example exists extension field fixed point formula Fourier geometry given graph Hilbert Hilbert space holomorphic homeomorphism infinite integers integral Introduced by Professor invariant isomorphic January 18 lattice Lie groups linear manifold Math Mathematical Reviews matrix maximal measure method metric Michigan minimal nonlinear obtained October 13 operators orthogonal polynomials paper polynomials Preliminary report prime problem proof properties prove Received August Received February Received January Received June Received October 13 Received September recursive representation ring satisfies semigroups semilattice sequence singular solutions structure subgroup subset subspace surface Theorem topological transform University vector zero